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Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals2 Marks
Question
Evaluate the following integrals:$\int\cos\sqrt{\text{x}}\text{dx}$
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Answer

Let $\text{I}=\int\cos\sqrt{\text{x}}\text{dx}$
$\sqrt{\text{x}}=\text{t}$
$\text{x}=\text{t}^2$
$\text{dx}=2\text{t dt}$
$=\int2\text{t}\cos\text{t dt}$
$\text{I}=2\int\text{t}\cos\text{t dt}$
$\text{I}=2[\text{t}\int\cos\text{t dt}-\int(1\int\cos\text{t dt})\text{dt}]$
$=2[\text{t}\sin\text{t}-\int\sin\text{t dt}]$
$=2[\text{t}\sin\text{t}+\cos\text{t}]+\text{C}$
$\text{I}=2[\sqrt{\text{x}}\sin\sqrt{\text{x}}+\cos\sqrt{\text{x}}]+\text{C}$

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